A new extended gumbel distribution: Properties and application

A robust generalisation of the Gumbel distribution is proposed in this article. This family of distributions is based on the T-X paradigm. From a list of special distributions that have evolved as a result of this family, three separate models are also mentioned in this article. A linear combination of generalised exponential distributions can be used to characterise the density of a new family, which is critical in assessing some of the family’s properties. The statistical features of this family are determined, including exact formulations for the quantile function, ordinary and incomplete moments, generating function, and order statistics. The model parameters are estimated using the maximum likelihood method. Further, one of the unique models has been systematically studied. Along with conventional skewness measures, MacGillivray skewness is also used to quantify the skewness measure. The new probability distribution also enables us to determine certain critical risk indicators, both numerically and graphically. We use a simulated assessment of the suggested distribution, as well as apply three real-world data sets in modelling the proposed model, in order to ensure its authenticity and superiority.

The EGTT model is right skewed with only one non-monotonic hazard rate shape (upside down bathtub). Two real life data sets have been analyzed with only two comparative models. However, the model presented in the newly proposed manuscript Equation 3, given as follows Out of so many, the following are some of the significant points which shows the uniqueness of the proposed model, i.e.
i. The proposed model discusses the Gumbel-Type I distribution in the context of extreme value theory. ii. The proposed model presents a generalization of Gumbel-Type I distribution, denoted by EGuG. iii. The proposed model presents the properties of generalized class. iv. The proposed model presents the functional forms of at least eleven special models, out of which the distribution function (cdf) and density functions (pdfs) of three models arising due to the cdf proposed in equation 3 are obtainable, which can be studied as a result of the proposed generalization. v. Out of the three presented special models, Nadarajah Haghighi distribution is taken as baseline and the model EGuNH has further been studied thoroughly. vi. The proposed special model outperforms five well established generalized classes of distribution along with the baseline model which solidifies its superiority over its existing generalizations.
b) In the article, cited as Hormatollah Pourreza et.al (2021), the authors proposed a generalization of two parameter Gamma distribution (Gamma-X) using the link function [ ( )] = − log[1 − ( )], following the methodology introduced by 2 Miroslav M. Ristic and Narayanaswamy Balakrishnan (2012) to propose the generalization of one parameter Gamma distribution and studied the Gamma exponentialted exponential (GEE) distribution. The cdf of Gamma-X family is given as Further, the authors took Weibull as baseline and studied the properties of Gamma-Weibull distribution. ii. The proposed generalization is useful to unbounded interval of data with range −∞ ≤ ≤ +∞ which is unique only to this model. iii. The proposed model discusses the Gumbel-Type I distribution in the context of extreme value theory with applications related to environmental and health hazards, an area which is under explored in the extreme value theory. iv. Nadarajah haghighi distribution is used as baseline modelwhich is unique for zero inflated models unlike Weibull distribution. v. The EGuNH model with only three parameters yields all four hazard rate shapes including monotone (Increasing & decreasing) and nonmonotone (bathtub & upside down bathtub) whereas Gamma-Weibull model yields only three shapes of hazard rate function (Increasing, decreasing & Bathtub) with four parameters.
Minor comments 1. In the Abstract, the authors used the term "greatest likelihood method". Is it the maximum likelihood method? If so, I think the common term should be the maximum likelihood method.
Reply: Thanks for pointing out; it was a typing error. It has been take care of.

In Section 1, the authors did not explain what T-X methodology is. It would be good if the authors can provide a citation.
Reply: Thank you for the valuable suggestion. T-X methodology is an integral part of modern distribution theory. As the article was already lengthy, we excluded it to keep the article length short and concise. However, we have now added a brief description and the citation of the T-X methodology paper in the manuscript.
3. In Section 2, it would be good to use the hazard rate function to define "hrf" rather than failure rate function.

Reply:
Thanks for pointing out; it has been dealt with as per suggestion.
4. In-Page 13, the authors did not cite figure # for "Plots of MGs for some parameter values".

Reply:
Thanks for pointing out; it has been corrected.
5. The tables should be self-explanatory. Please explain the acronyms in the tables.
For example, what is LoS in Table 12?

Reply:
Thanks for pointing out; it was a typing error. The acronym in the table has now been explained.
6. The authors seem did not explain most of figures and tables. For example, what is the message for Figure 13?
Reply: Risk measures are statistical indicators of the uncertainty of financial institutions or asset portfolios.